Built on Facts

From the inimitable xkcd, a comic about relativity in sex:

This rather clever piece of humor is referring to the relativity of simultaneity. According to relativity, time and space are different for observers in different reference frames. Events which are simultaneous in one frame are not necessarily going to be simultaneous in a different frame moving with respect to the first frame. This isn’t just a failure to correct for light travel time either, it’s a genuine geometric effect of spacetime.

So if one person is, er, moving at near the speed of light the effect can be quite pronounced. Simultaneous for one person might not be so simultaneous for another.

What is the same for both observers is the spacetime interval between two events (Let’s say the first event is their pre-coital synchronization of their watches, as any good experimentalists would surely do.) The spacetime interval s² is defined as the following*:

And the spacetime interval is invariant. For two different observers the time and position of an event may be different, but this combined quantity is necessarily going to be the same for both people. A change in the x, y, and z coordinates of an event (say, as a result of length contraction) is necessarily going to cause a shift of the t coordinate as well.

However, here the “event” in question happens at the same location for both people. The change in location is zero. Therefore the change in time is going to be zero as well. There’s no accumulated relativistic effects to damage the synchronization. Leaving aside the possibility of, uh… falsified data, I’d assume that in fact one of the experimenters messed up the initial clock synchronization. Perhaps forgivable, considering the level of distraction.

*The square of s is generally considered to be the fundamental quantity, not s itself.

Comments

#1 Arikia
December 12, 2008

Brilliant.
#2 Coturnix
December 12, 2008

Thank you, thank you! I was totally stumped by that XKCD cartoon….but now I know better
#3 Matt Heath
December 12, 2008

It has cool tooltip about the twin paradox as well.
#4 Uncle Al
December 12, 2008

Serves her right for not being in a co-moving frame. Was his performance time-like (teenager) or space-like (Viagra)?
#5 Abby Normal
December 12, 2008

A change in the x, y, and z coordinates of an event (say, as a result of length contraction) is necessarily going to cause a shift of the t coordinate as well.

I love it when you talk dirty.
#6 Robert
December 12, 2008

Only it’s spelled simultaneity.

Fixed. At least it was right in the text itself! -Matt
#7 Bill
December 12, 2008

Still, it is common for couples in Finland to finish simultaneously.
#8 andy.s
December 13, 2008

At those speeds, Lorentz contraction of the male’s, uh, “x-axis” makes you wonder, “why bother”?
#9 Paul Murray
December 14, 2008

I like to explain the relativity of simultaneity by pointing out that in space, *stationary* is relative, and noting that “simultaneous” is the same as “stationary”, but with the roles of space and time flipped around.

Of course this doesn’t mean that simultaneity is nessesarily relative, but observations confirm that this is, in fact, how rthe universe works.

Once we have established the relativity of simultaneiety, then the lenght paradox becomes explicable. To measure the length of something that is moving, you have to put a drawing pin into each end *at the same time*, and then measure the distance between those two stationary drawing pins. If two people disagree about simultaniety, then they won’t ever put their pins in at the same two points.

And so on.